What I would do: Add a separate distribution panel for the loads you want to feed from the inverter. Add an automatic transfer switch set up so that priority is the feed from the existing converter breaker - that way if there is power from the post or the generator it is first priority, and the inverter power is only switched on if there is no power from post or gennie. I would add 12VDC cables directly to the batteries for the input side of the inverter. Next I would upgrade the 12/2 cable from the post and the generator input to the converter - 12 gauge is only rated to 20 amps but you have a 30 amp breaker, so your are overloaded for the wire size. Or install a 20 amp main breaker. Be aware that only permanent connections to the inverter are code compliant in a permanent install - connecting with a plug is not compliant (but everyone does it anyway).
Brian